Scope and sequence
2008/19589[4]
Mathematics Specialist: Scope and sequence
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Mathematics Specialist: Scope and sequence
Content
Organization of content in the Mathematics: Specialist course
The course content comprises concepts and relationships for vectors, trigonometry,
exponentials/logarithms, functions, mathematical reasoning, complex numbers, polar
coordinates and matrices.
Polar coordinates are included only in units 3AMAS and 3CMAS, vectors in units 3AMAS,
3BMAS and 3CMAS and matrices in unit 3DMAS. The other content areas are included in all
units. The cognitive difficulty of the content increases across the units. There is a comprehensive
scope and sequence document which shows the progression in the content areas through the
four units.
Within the broad area of mathematical relationships, teachers may choose a variety of contexts
appropriate for the age group, interests and locality of the students.
Mathematics Specialist: Scope and sequence
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Mathematics Specialist: Scope and sequence
Mathematics: Specialist course (MAS)
Scope and Sequence
The syllabus content is presented here in a table format to assist teachers
to follow the order of content across units 3AMAS to 3DMAS
Mathematics Specialist: Scope and sequence
5
Unit 3AMAS
1. Vectors (11 hours)
1.1 distinguish between vector and
scalar quantities
1.2 represent a directed line segment
in the plane with magnitude and
direction using vector
1.3
1.4
1.5
1.6
displacement notation AB or a
develop the concept of equality of
vectors, opposite vectors, unit
vectors and the zero vector
represent a vector as an ordered
pair (a,b)
represent vectors in the form
ai + bj, where i and j are the
standard unit vectors
establish and use the formula
a, b 
a 2  b 2 for the
magnitude or modulus of a vector
as its length in the plane
1.7
1.8
define the position vector OP ,
from the origin, of a point P in the
Cartesian plane
use the parallelogram law or
triangle law of vector addition and
the triangle inequality
ab  a  b
1.9
multiply a vector by a scalar and
subdivide line segments internally
1.10 represent relative displacement
and relative velocity as vectors.
6
Unit 3BMAS
Unit 3CMAS
Unit 3DMAS
1. Vectors (16 hours)
1.1 develop the concept of the
dot product of vectors in a
plane, using projections,
and the formula
a  b  a1b1  a2b2 , and
establish the formula
a  b  a b cos where
1. Vectors (12 hours)
1.1 review vector properties in 2D and extend into
3D, namely: represent vectors in space in
Cartesian form as ordered triples (a,b,c)
1.2 develop the concept of displacement vectors
in space, including equality of vectors,
opposite vectors and the zero vector
1.3 establish and use the formula
1.2 calculate the angle
between two vectors and
identify parallelism and
perpendicularity
1.3 establish and use the
vector equation of a line in
the plane in its various
forms:
one point and slope:
vector in space
represent the vector (a,b,c) in the form
ai + bj + ck, where i, j and k are the
standard unit vectors
develop the concept of the position vector of a
point in space
add vectors in space using the parallelogram
rule and addition of components
multiply vectors by scalars and extend this to
subdividing line segments internally
develop the concept of the dot product of
vectors in a plane, using projections, and the
formula a  b  a1b1  a2b2  a3b3 and establish
1. Matrices (14 hours)
1.1 add, subtract and multiply
matrices (including multiply by a
scalar)
1.2 examine the algebraic properties
of matrix addition and
multiplication, including
commutativity for addition and
not for multiplication
1.3 examine the properties of special
matrices: identity, unit, singular,
diagonal, row and column
matrices
1.4 calculate the determinant and
inverse of a 2 x 2 matrix and
recognise a singular 2 x 2 matrix
1.5 solve systems of up to five
simultaneous linear equations
with no more than five
unknowns, using matrix algebra
1.6 examine the geometric properties
of 2 x 2 matrices as linear
transformations in the plane,
including general rotations and
reflections, and dilations and
shears parallel to the coordinate
axes
1.7 use matrix multiplication to
determine the combined effect of
two linear transformations in the
plane
1.8 establish and apply the
relationship between the
determinant and areas of shapes
before and after transformation
1.9 solve practical problems
involving the use of Leslie
matrices and other examples of
transition matrices.
a, b,c  
a  a1 ,a2  and b  b1 ,b2 
r = r1+l
1.4
1.5
1.6
1.7
1.8
two points:
r = r1 + (r2 – r1)
normal: r n  c
1.4 establish and use the
vector form of the equation
of a circle in the plane:
r-d  
a 2  b 2  c 2 for the length of a
the formula a  b  a b cos where
a  a1 ,a2 ,a3  and b  b1 ,b2 ,b3 
1.9
1.5 solve practical problems
using vector equations of
lines and the dot product
including tangency and
shortest distance
problems.
1.6 solve practical problems
using vectors including.
the study of bearings,
forces and navigation
problems involving
apparent and true
velocities.
calculate the angle between two vectors and
identify parallelism and perpendicularity
1.10 establish and use the vector equation of a
plane
1.11 establish and use the vector equation of a line
in space in the form r = r1+l together with
its parametric equivalent
1.12 solve practical problems in three-dimensional
geometry using vector concepts and formulas,
and graphical methods where appropriate.
Mathematics Specialist: Scope and sequence
Unit 3AMAS
Unit 3BMAS
2. Trigonometry (11 hours)
2.1
establish the relationship
between radian measure and
degree measure of angles
and convert from one
measure to the other
2.2
determine arc lengths in
circles, exactly and
approximately
2.3
establish and use the formula
for the area of triangles
1
area ΔABC = absin C
2
2. Trigonometry (10 hours)
2.1 develop the concept of sine, cosine and tangent as functions,
and establish and use the following properties:
2.4
2.5
2.6
2.7
determine areas of sectors
and segments in circles
using exact and approximate
values as appropriate
establish and use the sine
and cosine rules to find
distances and angles in
triangles in two and three
dimensional situations,
including obtuse triangles
and those triangles with two
solutions (the ambiguous
case)
use the triangle inequality for
the lengths of the sides of a
triangle
solve practical problems;
including angles of elevation
and depression, surveying,
bearings and navigation
distances along circles of
constant latitude or constant
longitude on the surface of
the Earth.
Unit 3CMAS


2  x
establish the limit
2.2
as x  0 using inequalities,
graphically and numerically
establish
the
periodicity:
sin x  2   sin x , cosx  2   cos x , tan x     tan x



phase: sin x  cos x  2 , cos x  sin x  2
2.2
2.3
2.4

investigate the transformations of sine, cosine and tangent
functions such as y = a sin b(x + c) + d and identify the
effects of the constants a, b, c and d on amplitude, period,
phase, and the locations of zeros and turning points (see
3AMAS 4.6)
use appropriate technology to investigate and represent
diagrammatically the roles of a, b, c and d in the linear scale
changes studied in 2.2 above
use the addition and double angle formulas for sine, cosine
and tangent:
sin      sin  cos  cos sin 
sin x
1
x
2.1
Pythagorean: sin 2 x  cos 2 x  1
parity: cos x   cos x , sin  x    sin x , tan  x    tan x
complementarity: sin x  cos 2  x ; cos x  sin
Unit 3DMAS
2. Trigonometry (6 hours)
limit
1  cos x
0
x
as
x 0
2.3
2.4
determine the derivative of
sin(x), from first principles
differentiate and integrate the
sine, cosine and tangent
functions.
2. Trigonometry (4 hrs)
2.1 investigate
the
differential equation
d2y
 k2y  0
dx 2
and its solutions
y (0)
sin kt
k
 A cos( kt   1 )
y(t )  y(0) cos kt 
 A sin( kt   2 )
as models of simple
harmonic motion.
cos     cos cos  sin  sin 
tan   tan 
1  tan  tan 
sin 2  2 sin  cos
cos 2  cos2   sin 2 
tan     
 2 cos2   1
 1  2 sin 2 
2 tan 
tan 2 
1  tan 2 
sin2θ  2sinθ cos
cos2θ  cos 2θ  sin 2θ
 2cos 2θ  1  1  2sin 2θ
2.5
solve trigonometric equations of the form
sin(ax) = k, cos(ax) = k and tan(ax) = k for a given finite
domain.
Mathematics Specialist: Scope and sequence
7
Unit 3AMAS
3. Exponentials and logarithms
(13 hours)
3.1 develop and use the index laws for
positive bases and rational
exponents
3.2 establish and use the properties of
exponential functions
y  Ca x a  0 and draw their
graphs
3.3 develop the inverse relationship
between logarithmic and
exponential functions:
x  log y and
3. Exponentials and logarithms
(8 hours)
3.1
investigate the limiting
behaviour of
r n as n  , ( r  1)
3.2
4. Functions (13 hours)
4.1
develop the concept of function
composition and obtain
expressions for the composites of
simple functions
4.2
identify the domain and range of
simple functions and their
composites
4.3
investigate the inverse of a
function as a reflection in y = x
4.4
investigate relationships between
domains and ranges of functions
and their inverses
4.5
solve, algebraically and
geometrically, simple equations
and inequalities involving absolute
investigate the limiting
behaviour of
a

1  
n

3.3
3.4
3.5
Unit 3CMAS
3. Exponentials and logarithms
(7 hours)
3.1
review the inverse relationship
between exponentials and
logarithms
3.2
investigate the logarithmic
properties of the function
1
1 t dt , define this as the
natural logarithm ln x and
x
n
as n   , (a fixed)
define e as the limit of
1
y  ax
a
3.4 investigate and use the properties of
the logarithmic functions
y  log x for a > 0, and draw their
a
graphs
3.5 use the laws of logarithms
3.6 solve growth and decay problems
using exponential and logarithmic
functions.
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Unit 3BMAS
n

 1   as n  
n

investigate growth and decay
problems of the form y = a.ekx
differentiate exponential and
logarithmic functions including
ef(x) and ln[f(x)].
4. Functions (12 hours)
4.1
investigate the continuity and
limiting behaviour of functions
4.2
define the derivative of functions
from first principles and apply to
familiar functions (not
trigonometric)
4.3
investigate the differentiability of
functions using limits
4.4
draw and interpret graphs of
gradient functions
4.5
investigate piecewise-defined
functions and continuity (including
absolute value, the sign function
sgn [x], and the greatest integer
function int [x] )
3.3
3.4
Unit 3DMAS
3. Exponentials and logarithms
(7 hours)
3.1 integrate linear combinations of
powers and exponentials
3.2 solve practical problems involving
models of growth and decay of the
form
dP
 kP
dt
3.3 solve practical problems involving
logarithmic scales.
review its basic properties
use the change of base formula
to convert logarithms from one
base to another
integrate functions of the
form
kf x 
f x 
and kf x e
f x 
using the change of variable (or
substitution technique) either by
observation, or provided.
4. Functions (16 hours)
4.1 determine the derivative of
polynomial functions from first
principles
4.2 use the product, quotient and chain
rules to differentiate functions
including exponential, logarithmic
and trigonometric functions
4.3 find the area under and between
curves
4.4 determine the equation(s) of the
tangent(s) to a function
4.5 differentiate functions defined
implicitly
4.6 solve practical problems involving
parametric
and
differential
Mathematics Specialist: Scope and sequence
4. Functions (10 hours)
4.1 investigate graphical, geometric and
algebraic properties of absolute
value functions (in the complex and
Cartesian planes)
4.2 integrate functions and composite
integrands involving power,
polynomial, exponential, logarithmic
and trigonometric functions studied,
using the change of variable (or
substitution technique) either by
observation, or as provided
4.3 solve related rates problems
(including those involving
trigonometric functions)
4.4 solve practical problems by applying
Unit 3AMAS
4.6
values of linear functions
investigate the effects of varying
a, b, c and d on the graph of
y  af b( x  c)  d where



f x is an exponential,
logarithmic, power, reciprocal or
absolute value function.
5. Mathematical reasoning
(3 hours)
5.1 identify and generalise number
patterns for powers, exponential,
and inverse relationships
5.2 establish the laws of logarithms
5.3 investigate properties of the
absolute value function (real)—
analytically and graphically.
Unit 3BMAS
4.6
4.7
4.8
apply the chain rule with
appropriate notation to
differentiate composite functions
use the product and quotient rules
to differentiate polynomial,
exponential (base e) and natural
logarithmic functions.
develop the concept of the integral
of a function as a limiting sum
5. Mathematical reasoning
(4 hours)
5.1 make conjectures regarding limiting
patterns
5.2 establish the addition and double
angle formulas for sine, cosine and
tangent
5.3 develop the chain rule for
differentiating composite functions
5.4 prove simple trigonometric identities
by deduction, using the properties
listed in 2.1 and 2.4 above
Unit 3CMAS
Unit 3DMAS
equations (variables separable)
integrate
combinations
of
functions using antiderivatives
integrate functions using the
change of variable or substitution
technique (either by observation, or
provided).
calculus techniques to problems
from various branches of the
sciences including rectilinear motion
and marginal cost.
5. Mathematical reasoning
(5 hours)
5.1 make conjectures and
generalisations about properties of
natural and figurate numbers and
recurrence relations
5.2 find counterexamples to disprove
mathematical statements
5.3 distinguish between axioms and
theorems
5.4 develop geometric proofs by
deduction using vector methods.
5.5 prove harder trigonometric
identities by deduction, using the
properties in 3BMAS 2.1 and 2.4
5.6 explore proof by exhaustion
5.7 explore proof by contradiction
including Euclid’s proof of ‘infinitely
many primes’
5. Mathematical reasoning
(5 hours)
5.1
investigate a variety of traditional
mathematical conjectures
including Goldbach’s conjecture
about two primes and the twin
prime conjecture.
5.2 explore proof by induction
including de Moivre’s theorem:
4.7
4.8
Mathematics Specialist: Scope and sequence
 z cis n  z n cis n
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6. Complex numbers (0 hours)
6.1
6.2
6.3
6.4
6.5
6.6
6. Complex numbers (5 hours)
develop the concept of the
number i as a solution of x2 = −1
investigate complex solutions of
quadratic equations
represent geometrically a complex
number z as a point in the
complex plane
represent the Cartesian form of z
as the sum of its real and
imaginary parts: z = a + bi,
where i2 = −1
add, subtract, multiply and divide
complex numbers in Cartesian
form
develop the concept of conjugates
of complex numbers.
6. Complex numbers (7 hours)
6.1
express a complex number z in
polar form:
z = r cis , where r = z and
 = arg z
6.2
multiply and divide complex
numbers expressed in polar form
6.3
6.4
2
z z  z ; z1  z 2  z1  z 2 ; z1 z 2  z1 z 2
6.5
6.6
7. Polar coordinates (2 hours)
7.1
develop the concept of polar
coordinates (r, ) in the plane,
where r  0
7.2
use the relationship between
Cartesian and polar coordinates
in the plane and convert from
one system to the other.
10
determine the conjugate z of a
complex number z, expressed in
Cartesian or polar form, and locate
it in the complex plane
establish algebraically and
geometrically, the conjugation
properties:
z
establish
z
2
as the reciprocal
of a non-zero complex number, z
describe regions in the complex
plane and Argand diagrams
defined by means of simple
systems of equalities and
inequalities.
7. Polar coordinates (2 hours)
7.1 find the distance between points
whose position is expressed in
polar form
7.2 draw and interpret polar graphs
(including inequalities) of
r = constant, θ = constant and
r =k θ
Mathematics Specialist: Scope and sequence
6. Complex numbers (12 hours)
6.1 establish properties of sums,
products, division and
exponentiation (including
combinations of these) of complex
numbers and their conjugates
(using real and ‘imaginary’
components)
6.2 use de Moivre’s theorem
 z cis n  z n cis n
6.3
to establish
trigonometric relationships
find and locate in the complex
plane, solutions of z  C
establish the exponential
properties of cis   cos  i sin 
and use Euler’s formula
n
6.4
e i  cos  i sin  to investigate
cis    , cis 0 and cis  n  .